Orthogonal Matrices: Questions & Answers

[war485]

Homework Statement

1. If I got a square orthogonal matrix, then if I make up a new matrix from that by rearranging its rows, then will it also be orthogonal?

2. True/false: a square matrix is orthogonal if and only if its determinant is equal to + or - 1

Homework Equations

no equations

The Attempt at a Solution

1. I think it should also be orthogonal since it forms a basis, and the basis would be the same, but just a linear combination of the previous, right?

2. false, its determinant doesn't necessarily ensure it is orthogonal. So, how would/should I correct that statement?

 

[NoMoreExams]

You are right about #2: http://en.wikipedia.org/wiki/Orthogonal_matrix#Matrix_properties

 

[Unco]

1. If I got a square orthogonal matrix, then if I make up a new matrix from that by rearranging its rows, then will it also be orthogonal?

1. I think it should also be orthogonal since it forms a basis, and the basis would be the same, but just a linear combination of the previous, right?

An nxn matrix is orthogonal iff its rows form an orthormal basis for \mathbb{R}^n (note the symmetry of AA^T=A^TA=I for an orthogonal matrix A). The linear independence of a collection of vectors doesn't depend on the order in which you write them, so the rows of the new matrix still form an orthonormal basis.

Just be careful your language: a linear combination of a basis reads as a linear combination of its vectors, which gives just one vector.

 

[war485]

I forgot about the linear independence part of it for #1.

As for #2, I took a counter-example from wikipedia XD
[ 2 0 ]
[ 0 0.5 ]
where its determinant = 1
but the length of each column is not 1 (not orthonormal)
I guess counter-examples should be enough?

Thanks for the help you two. :)

 

[Unco]

As for #2, I took a counter-example from wikipedia XD
[ 2 0 ]
[ 0 0.5 ]
where its determinant = 1
but the length of each column is not 1 (not orthonormal)
I guess counter-examples should be enough?

The statement #2 is (colloquially) of the form "(property X implies property Y) AND (property Y implies property X)" (*). If all you want to do is show that (*) is false (e.g., if you were asked to prove or disprove the statement), then it suffices to show that property Y does not imply property X.

To show that property Y does not imply property X, it suffices to give an example for which property Y holds but X does not. Why? Because it definitively answers the question as to whether Y implies X. There is no guessing about it!

 

[war485]

Yea, you're right Unco. I need to work on my logic a bit more. I'm very grateful for your help :D